To express the formula you provided in standard notation for a geometric sequence, let's first clarify the notation.
The expression \( a_n = 3 \cdot 9^{(n-1)} \) represents the \( n \)-th term of a geometric sequence where:
- The first term (\( a_1 \)) occurs when \( n = 1 \).
- The common ratio (\( r \)) is derived from the base of the exponent, which is 9 in this case.
Step 1: Calculate the first term \( a_1 \)
Substituting \( n = 1 \): \[ a_1 = 3 \cdot 9^{(1-1)} = 3 \cdot 9^0 = 3 \cdot 1 = 3 \]
Step 2: Calculate the common ratio \( r \)
The common ratio \( r \) can be determined by dividing the second term by the first term or recognizing it from the structure: \[ r = 9 \quad \text{(as it's the base of the exponent)} \]
Step 3: Write the general form of the geometric sequence
A geometric sequence can be represented generally as: \[ a_n = a_1 \cdot r^{(n-1)} \] Substituting the known values:
- \( a_1 = 3 \)
- \( r = 9 \)
The sequence can be written as: \[ a_n = 3 \cdot 9^{(n-1)} \]
This confirms that the terms of your sequence are of the form \( 3, 27, 243, \ldots \) as you calculate \( a_2, a_3, \) etc. explicitly.
Conclusion:
Thus, \( 3 \cdot 9^{(n-1)} \) is already in the standard form of a geometric sequence where the first term is 3 and the common ratio is 9.
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